Moments
Can uniquely characterize a distribution
Central Moments
Central Moments: \(\mu_r=\frac{\sum(x_i-\bar x)^r}{n}\)
- \(\mu_1=\frac{\sum(x_i-\bar x)}{n}=\frac{\sum
x_i}{n}-\frac{n \bar x}{n}=\frac{n \bar x}{n}-\frac{n \bar
x}{n}=0\)
- \(\mu_2=\frac{\sum(x_i-\bar
x)^2}{n}=\sigma^2\)
- \(\mu_3=\frac{\sum(x_i-\bar x)^3}{n}\)
- \(\mu_4=\frac{\sum(x_i-\bar x)^4}{n}\)
- For grouped data: \(\mu_r=\frac{\sum f_i(x_i-\bar
x)^r}{n}\)
Raw Moments
\(\mu_r'=\frac{\sum(x_i-a)^r}{n}\); a is
arbitrary number
\(\mu_1'=\frac{\sum(x_i-a)}{n}=\frac{\sum
x_i}{n}-\frac{na}{n}=\bar x-a\)
Value of 1st Central Moment
\(\displaystyle \mu_1=\frac{\sum (x_i-\bar
x)}{n}=\frac{\sum x_i}{n}=\bar x - \bar x = 0\)
Changing Origin of Moments
From \(\mu_r'(a)\) to \(\mu_r'(k)\)
Assume, \(a^r = \mu_r'(a)\),
\(b = a - k\)
Binomial Formulae
- \((a+b)^1 = a + b\)
- \((a+b)^2 = a^2 + 2ab + b^2\)
- \((a+b)^3=a^3+3a^2b+3ab^2+b^3\)
- \((a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4\)
- \(\mu_1'(k) =
\mu_1'(a) + b\)
- \(\mu_2'(k) =
\mu_2'(a) + 2 \mu_1'(a)b + b^2\)
- ?
- ?
See Examples
Central Moments from Raws
Use \(k = \bar x\)
That’s it! DO NOT MEMORIZE!
2nd Central Moment
\(\mu_2=\mu_2' -
\mu_1'^2\)
- \(\mu_2=\mu_2'(a)+2\mu_1'(a)(a-\bar
x)+(a-\bar x)^2\) [From here]
- \(\mu_2'-2(a-\bar
x)^2+(a-\bar x)^2\) [since \(\mu_1'(a) = \bar x-a\)]
- \(\mu_2'-(a-\bar
x)^2\)
- \(\mu_2' -
\mu_1'^2\)
Chracteristics of Moments
- Used for testing for symmetry, normality, and
skewness
- First raw moment around zero is Arithmetic Mean
(\(\mu_1'=\frac{\sum (x_i)}{n}=\bar
x\))
- Second central moment is equal to variance (\(\sigma^2 = \frac{\sum (x_i-\bar x)^2
)}{n}\))
- 2nd and 3rd central moments are used to measure
skewness (detailed later)
- 2nd and 4th central moments are used to measure
kurtosis
Can Moments Be Negative?
\(\displaystyle \mu_r = \frac{\sum(x_i-\bar
x)^r )}{n}; r=1,2,3, \cdots\)
- 0 if all values are equal to origin
- Negative if r is an Odd number
- Positive if r is Even number
Skewness
Lack of symmetry
- +Ve Skew \(\rightarrow \bar X \gt Me \gt
Mode\)
- -Ve Skew \(\rightarrow \bar X \lt Me \lt
Mode\)
- No Skew \(\rightarrow \bar X = Me =
Mode\)
Positive Skewness
- Low values have high frequency
- High values have low frequency
- As the value increases, the frequency decreases
Most students score low, very few get good scores.
Negative Skewness
- Low values have low frequency
- High values have high frequency
- As the value increases, so does the frequency
Very few students get low marks, most get high marks.
Symmetric
- Average values have maximum frequency
- Low values have low frequency and high values have high
frequency
- As the value go farther from mean, their frequency gradually
decreases.
Kurtosis
Normal distribution \(\rightarrow\)
- Most students get average marks
- Higher and lower marks are obtained by lesser number of
students
Kurtosis Misconception
Kurtosis is NOT a degree of peakedness. Learn
more
Measures of SKewness
- Bowley’s Coefficient: \(SK_B=\frac{Q_3+Q_1-2Me}{Q_3-Q_1};
(-1,1)\)
- Kelly’s Coefficient: \(SK_k=\frac{D_1+D_9-2Me}{D_9-D1}\)
- Method of Moments: \(\beta_1=\frac{\mu_3^2}{\mu_2^3}\)
Estimate Skewness
4, 23, 55, 70, 74, 78, 86, 89
Measures of Kurtosis
- Pearsons’s Coefficient of Moments, \(\beta_2 = \frac{\mu_4}{\mu_2^2}\)
- Percentile Coefficient, \(K=\frac{\frac 1
2 (Q_3-Q_1)}{P_{90}-P_{10}}\)
Five Number Summary
- Minimum
- Maximum
- Quartiles: \(Q_1, Q_2, Q_3\)
- \(Q_1 - X_L<X_H-Q_3
\rightarrow\) Positive Skew
- \(Q_1 - X_L>X_H-Q_3
\rightarrow\) Negative Skew
Example of Five Number Summary
2, 1, 0, 5, -6, 7, -4
Solution
Box and Whisker Plot
